If $f(x) = \begin{cases} xe^{-\begin{bmatrix}\frac{1}{|x|} - \frac{1}{x}\end{bmatrix}}, & \quad \text{$x$ $\neq$ 0} \text{ then $f(x)$ is :}\\ \; \; \; 0 & \quad \text{$x=0$} \end{cases}$

Question

( A ) continuous as well as differentiable for all $x$
( B ) continuous for all $x$ but not differentiable at $x=0$
( C ) discontinuous everywhere
( D ) neither differentiable nor continuous at $x=0$